2. Description
① During the whole cycle, after a series of state changes, the internal energy of an ideal gas remains unchanged, but it needs work and heat exchange. The cycle is divided into four processes. It is represented by two isotherms and two adiabatic lines on the p-V diagram (as shown in the figure). In the figure, curves AB and CD are two isotherms with temperatures of T 1 and T2, and curves BC and DA are two adiabatic lines. We discuss the circulation along the closed curve ABCDA clockwise on the p-V diagram. (This cycle is called positive cycle of working medium. A machine that performs a positive cycle is called a heat engine. This is a machine that converts heat into work. )
The first process: A→B, isothermal expansion, q1= EB-ea+w1;
The second process: B→C, adiabatic expansion, O = EC-e B+W2；;
The third process: C→D isothermal compression,-Q2 = ED-EC-W3;
The fourth process: D→A, adiabatic compression, O=EA-ED-W4.
Add the above four formulas to get q1-Q2 = w1+w2-w3-w4 = w0.
Where q is the heat absorbed from the high-temperature heat source, Q2 is the heat released to the low-temperature heat source, and W is the net work done by the ideal gas (working medium), which is numerically equal to the area surrounded by the closed curve on the p-V diagram.
Q 1-Q2=W .
The above formula shows that the heat Q 1 absorbed by an ideal gas from a high-temperature heat source does external work, and the other part is released to a low-temperature heat source (as shown in the figure). That is to say, heat Q 1 cannot be completely converted into work w, only Q 1-Q2 can be converted into work. The thermal efficiency of a general heat engine is expressed as η t = w/q1= (q1-Q2)/q1=1/Q2.
Since Q2 cannot be equal to zero, the thermal efficiency of a heat engine is always less than L, and ηt is usually expressed as a percentage.
② Carnot further proves theoretically that in Carnot cycle,
Heat absorbed during isothermal expansion QL = nrtl1nv2/v1(1).
Heat released during isothermal compression Q2=nRT2lnV3/V4, (2)
T1Tv2γ-1= T2TV3γ-1(3) can be obtained from the adiabatic equation TVγ- 1= constant.
t 1 TV 1γ- 1 = T2 TV 4γ- 1(4)
Where t represents the absolute temperature of high-temperature heat source and t represents the absolute temperature of low-temperature heat source.
The formula shows that all heat engines must have two heat sources: high temperature and low temperature to complete a cycle. The thermal efficiency of the heat engine is only related to the temperatures of the two heat sources, and has nothing to do with the working medium. The greater the temperature difference between the two heat sources, the higher the thermal efficiency, that is, the greater the utilization rate of heat absorbed from the heat sources. To improve thermal efficiency, it is necessary to raise the temperature of high-temperature heat source or lower the temperature of low-temperature heat source. Generally take the former. This formula points out a way to improve the efficiency of heat engine for people.
③ Carnot cycle can also be carried out along the closed curve ADCBA in the counterclockwise direction of p-V diagram, which is called reverse cycle. In this reverse cycle, the outside world must do work for this system that absorbs heat from a low-temperature heat source. As long as the reverse cycle is repeated, any amount of heat can be taken away from the low-temperature heat source. The reverse circulation machine is called refrigerator, which uses external work to obtain low temperature.
Inverse Carnot cycle
It includes two isothermal processes and two adiabatic processes. Assuming that the temperature of the low-temperature heat source (i.e. the cooled object) is T0 and the temperature of the high-temperature heat source (i.e. the environmental medium) is Tk, then the temperature of the working medium
T0 in the process of heat absorption and Tk in the process of heat release, that is, there is no temperature difference between working medium, cold source and high-temperature heat source in the process of heat absorption and heat release, that is, heat transfer is carried out at isothermal temperature, and there is no loss in the process of compression and expansion. The circulation process is as follows:
First, the working medium absorbs heat q0 from the cold source (i.e. the object to be cooled) at time T0, and performs isothermal expansion of 4- 1, then increases its temperature from T0 to the temperature Tk of the environmental medium through adiabatic compression of 1-2, and then performs isothermal compression of 2-3 at time Tk, releasing heat qk to the environmental medium (i.e. the high-temperature heat source), and finally performs adiabatic expansion of 3-4.
For the inverse Carnot cycle, as can be seen from the figure:
q0=T0(S 1-S4)
qk=Tk(S2-S3)=Tk(S 1-S4)
w0 = qk-Q0 = Tk(s 1-S4)-T0(s 1-S4)=(Tk-T0)(s 1-S4)
Inverse Carnot cycle refrigeration coefficient εk is:
As can be seen from the above formula, the refrigeration coefficient of the inverse Carnot cycle has nothing to do with the nature of the working medium, but only depends on the temperature T0 of the cold source (i.e. the cooled object) and the temperature tk of the heat source (i.e. the environmental medium); Reducing Tk and increasing T0 can improve the refrigeration coefficient. In addition, the second law of thermodynamics can be used to prove that "the refrigeration coefficient of inverse Carnot cycle is the highest in a given temperature range of cold and heat sources". The refrigeration coefficient of any actual refrigeration cycle is less than that of the inverse Carnot cycle.
Generally speaking, the ideal refrigeration cycle should be the inverse Carnot cycle. In fact, the inverse Carnot cycle cannot be realized, but it can be used as an index to evaluate the perfection of the actual refrigeration cycle. Generally, the ratio of the refrigeration coefficient ε of the actual refrigeration cycle working at the same temperature to the refrigeration coefficient εk of the inverse Carnot cycle is called the thermal perfection of the refrigerator cycle, which is represented by the symbol η. Namely: η=ε/εk
Thermal perfection is used to indicate the degree to which the refrigerator cycle approaches the inverse Carnot cycle. It is also a technical and economic index of refrigeration cycle, but its meaning is different from that of refrigeration coefficient. For refrigerator cycles with different working temperatures, the economy of the cycle cannot be compared according to its refrigeration coefficient, but only according to its thermal perfection.